In
mathematics, particularly in
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, a webbed space is a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
designed with the goal of allowing the results of the
open mapping theorem and the
closed graph theorem
In mathematics, the closed graph theorem may refer to one of several basic results characterizing continuous functions in terms of their graphs.
Each gives conditions when functions with closed graphs are necessarily continuous.
Graphs and m ...
to hold for a wider class of
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s whose codomains are webbed spaces. A space is called webbed if there exists a collection of
sets, called a ''web'' that satisfies certain properties. Webs were first investigated by de Wilde.
Web
Let
be a
Hausdorff locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ...
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
. A is a stratified collection of
disk
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector space ...
s satisfying the following absorbency and convergence requirements.
# Stratum 1: The first stratum must consist of a sequence
of
disk
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector space ...
s in
such that their union
absorbs
# Stratum 2: For each disk
in the first stratum, there must exists a sequence
of disks in
such that for every
:
and
absorbs The sets
will form the second stratum.
# Stratum 3: To each disk
in the second stratum, assign another sequence
of disks in
satisfying analogously defined properties; explicitly, this means that for every
:
and
absorbs The sets
form the third stratum.
Continue this process to define strata
That is, use induction to define stratum
in terms of stratum
A is a sequence of disks, with the first disk being selected from the first stratum, say
and the second being selected from the sequence that was associated with
and so on. We also require that if a sequence of vectors
is selected from a strand (with
belonging to the first disk in the strand,
belonging to the second, and so on) then the series
converges.
A
Hausdorff locally convex topological vector space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topologica ...
on which a web can be defined is called a .
Examples and sufficient conditions
All of the following spaces are webbed:
*
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces ( normed vector spaces that are complete with respect ...
s.
*
Projective limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ...
s and
inductive limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categ ...
s of sequences of webbed spaces.
* A sequentially closed vector subspace of a webbed space.
* Countable
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Prod ...
s of webbed spaces.
* A Hausdorff quotient of a webbed space.
* The image of a webbed space under a
sequentially continuous
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
linear map if that image is
Hausdorff.
* The
bornologification of a webbed space.
* The continuous dual space of a metrizable locally convex space endowed with the
strong dual topology
In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded s ...
is webbed.
* If
is the
strict inductive limit of a denumerable family of locally convex metrizable spaces, then the
continuous dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by con ...
of
with the
strong topology In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to:
* the final topology on the disjoint union
* the ...
is webbed.
** So in particular, the
strong duals of locally convex
metrizable space
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty ...
s are webbed.
* If
is a webbed space, then any Hausdorff locally convex topology weaker than this (webbed) topology is also webbed.
Theorems
If the spaces are not locally convex, then there is a notion of web where the requirement of being a disk is replaced by the requirement of being
balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ci ...
. For such a notion of web we have the following results:
See also
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Citations
References
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{{Topological vector spaces
Functional analysis
Topological vector spaces